From the above, we see that reference frame transformations
in quantum mechanics are symmetries of the phase space. For
the Lie group theoretical ones as presented in Table 1. We have,
including the classical spatial or momentum translations, being
given in terms of one-parameter groups of Hamiltonian flows
which correspond to automorphism flows on the observable
algebra preserving the Poisson bracket. A `classical' spatial
translation as given by the unitary operator \(e^{ia\hat{p}}\) 
for a fixed real number \(a\)  is indeed, like its correspondence
in classical mechanics, a canonical transformation. However,
it should be emphasized that the real variable \(x\)  is not
a spatial coordinate for quantum mechanics. A spatial coordinate
should be an observable and a coordinate to the phase space as
a representation of the relativity symmetry. The true spatial
coordinate is exact the quantum counterpart of the classical \(x\) ,
i.e. \(\hat{x}\) . The `classical' spatial translation takes
\(\hat{x}\)  to \(\hat{x}-a\) , with the noncommutative value for
each state having only the first term, as the expectation value,
in the sequence representation shifted by \(a\) , hence the
quantum fluctuations are unchanged. The action on the
wavefunction \(\psi (x)\)  is the action on the latter as a set of
complex number coordinates of the phase space. The quantum
spatial translation considered above shifts the noncommutative
value \([\hat{x}_{\!\scriptscriptstyle C}]_{\scriptscriptstyle \phi }\)  for each `state of particle
\(C\) ' by a noncommutative value which is taken as the
\([\hat{x}_{\!\scriptscriptstyle B}]_{\scriptscriptstyle \phi }\)  with \(B\)  as the new reference
frame. However, the state involved is really a state of at least
the composite system of \(B\)  and \(C\) . With entanglement, not
only that we cannot talk about a state of \(C\)  in itself, we cannot
even talk about a generic state of the system with a fixed
\([\hat{x}_{\!\scriptscriptstyle B}]_{\scriptscriptstyle \phi }\) . The transformation hence can
only be given as a single unitary operator without any parameter
dependence or a notion of generator. Combining such
transformations is not a problem. It is straightforward to
check that two successive quantum spatial reference frame
transformations \(A \rightarrow B \rightarrow D\)  gives exactly the transformation
\(A \rightarrow D\)  for example, though one has to enlarge the composite
system to consider more and more parts [1]}, [2]}, [3]}. We have
seen also that in the special case with no entanglement in both
the initial and final states involved, such a description is admissible.
But it looks like the only such case has \(B\)  in a position
eigenstate. The case taken with the eigenvalue as a real variable
parameter matches exactly to the `classical' spatial translation
when effect of the transformation on the new nontrivial state
description of the old reference frame as a system is neglected.
